Edge-ordered Ramsey numbers

We introduce and study a variant of Ramsey numbers for edge-ordered graphs, that is, graphs with linearly ordered sets of edges. The edge-ordered Ramsey number $\overline{R}_e(\mathfrak{G})$ of an edge-ordered graph $\mathfrak{G}$ is the minimum positive integer $N$ such that there exists an edge-ordered complete graph $\mathfrak{K}_N$ on $N$ vertices such that every 2-coloring of the edges of $\mathfrak{K}_N$ contains a monochromatic copy of $\mathfrak{G}$ as an edge-ordered subgraph of $\mathfrak{K}_N$. We prove that the edge-ordered Ramsey number $\overline{R}_e(\mathfrak{G})$ is finite for every edge-ordered graph $\mathfrak{G}$ and we obtain better estimates for special classes of edge-ordered graphs. In particular, we prove $\overline{R}_e(\mathfrak{G}) \leq 2^{O(n^3\log{n})}$ for every bipartite edge-ordered graph $\mathfrak{G}$ on $n$ vertices. We also introduce a natural class of edge-orderings, called lexicographic edge-orderings, for which we can prove much better upper bounds on the corresponding edge-ordered Ramsey numbers.Comment: Minor revision, 16 pages, 1 figure. An extended abstract of this paper will appeared in the Eurocomb 2019 proceedings in Acta Mathematica Universitatis Comenianae. The paper has been accepted to the European Journal of Combinatoric

Erdős-Hajnal-type results for monotone paths

An ordered graph is a graph with a linear ordering on its vertex set. We prove that for every positive integer k, there exists a constant ck > 0 such that any ordered graph G on n vertices with the property that neither G nor its complement contains an induced monotone path of size k, has either a clique or an independent set of size at least n^ck . This strengthens a result of Bousquet, Lagoutte, and Thomassé, who proved the analogous result for unordered graphs. A key idea of the above paper was to show that any unordered graph on n vertices that does not contain an induced path of size k, and whose maximum degree is at most c(k)n for some small c(k) > 0, contains two disjoint linear size subsets with no edge between them. This approach fails for ordered graphs, because the analogous statement is false for k ≥ 3, by a construction of Fox. We provide some further examples showing that this statement also fails for ordered graphs avoiding other ordered trees

Stage-graph representations

AbstractWe consider graph applications of the well-known paradigm “killing two birds with one stone”. In the plane, this gives rise to a stage graph as follows: vertices are the points, and u, v is an edge if and only if the (infinite, straight) line segment joining u to v intersects the stage. Such graphs are shown to be comparability graphs of ordered sets of dimension 2. Similar graphs can be constructed when we have a fixed number k of stages on the plane. In this case, u, v is an edge if and only if the (straight) line segment uv intersects one of the k stages. In this paper, we study stage representations of stage graphs and give upper and lower bounds on the number of stages needed to represent a graph

Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs

A linearly ordered (LO) k-colouring of a hypergraph is a colouring of its vertices with colours 1, … , k such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO k-colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the "linearly ordered chromatic number" of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO 3-colourable, and the case that it is not even LO 4-colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opršal, Wrochna, and Živný (2023)

Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs

A linearly ordered (LO) $k$-colouring of a hypergraph is a colouring of its vertices with colours $1, \dots, k$ such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO $k$-colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the `linearly ordered chromatic number' of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO $3$-colourable, and the case that it is not even LO $4$-colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opr\v{s}al, Wrochna, and \v{Z}ivn\'y (2023)

Algorithms to Find Linear Geodetic Numbers and Linear Edge Geodetic Numbers in Graphs

Given two vertices u and v of a connected graph G=(V, E), the closed interval I[u, v] is that set of all vertices lying in some u-v geodesic in G. A subset of V(G) S={v1,v2,v3,….,vk} is a linear geodetic set or sequential geodetic set if each vertex x of G lies on a vi – vi+1 geodesic where 1 ? i < k . A linear geodetic set of minimum cardinality in G is called as linear geodetic number lgn(G) or sequential geodetic number sgn(G). Similarly, an ordered set S={v1,v2,v3,….,vk} is a linear edge geodetic set if for each edge e = xy in G, there exists an index i, 1 ? i < k such that e lies on a vi – vi+1 geodesic in G. The cardinality of the minimum linear edge geodetic set is the linear edge geodetic number of G denoted by legn(G). The purpose of this paper is to introduce algorithms using dynamic programming concept to find minimum linear geodetic set and thereby linear geodetic number and linear edge geodetic set and number in connected graphs